Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Fibonacci’s rabbits visit
the Mad Veterinarian
Gene Abrams
Pacific Lutheran University
Mathematics Colloquium
May 21, 2014
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Overview
1 Introduction and brief history
2 Mad Vet groups
3 Here’s where Fibonacci comes in ...
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
1 Introduction and brief history
2 Mad Vet groups
3 Here’s where Fibonacci comes in ...
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something familiar: The Fibonacci Sequence F (n)
Fibonacci’s Rabbit Puzzle: (from Liber Abaci, 1202)
Suppose you go to an uninhabited island with a pair of newborn
rabbits (one male and one female), who:
1
mature at the age of one month,
2
have two offspring (one male and one female) each month
after that, and
3
live forever.
Each pair of rabbits mature in one month and then produce a pair
of newborns at the beginning of every following month. How many
pairs of rabbits will there be in a year?
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something familiar: The Fibonacci Sequence F (n)
start month
n
1
2
3
4
5
6
7
8
9
10
11
12
···
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something familiar: The Fibonacci Sequence F (n)
start month
F (n)
n
1
2
3
4
5
6
7
8
9
10
11
12
1
1
2
3
5
8
13
21
34
55
89
144
···
···
There is a “generating formula” for the Fibonacci sequence:
F (1) = 1; F (2) = 1; F (n) = F (n − 1) + F (n − 2) for all n ≥ 3.
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something familiar: The Fibonacci Sequence F (n)
The Fibonacci sequence comes up in lots of places ...
AND is VERY well-studied!
For instance,
Theorem:
g.c.d.(F (n), F (m)) = F (g.c.d.(n, m)).
Corollary:
g.c.d.(F (n), F (n - 1)) = 1.
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something familiar: The Fibonacci Sequence F (n)
The Fibonacci sequence comes up in lots of places ...
AND is VERY well-studied!
For instance,
Theorem:
g.c.d.(F (n), F (m)) = F (g.c.d.(n, m)).
Corollary:
g.c.d.(F (n), F (n - 1)) = 1.
A site for all types of info about the Fibonacci sequence (more
than 300 formulas):
Google:
Ron Knott Fibonacci
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something not-as-familiar: Mad Vet Puzzles
Mad Vet Bob’s Mad Vet Puzzle: (from The Internet, 1998)
A mad veterinarian has created three animal transmogrifying
machines.
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Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something not-as-familiar: Mad Vet Puzzles
Mad Vet Bob’s Mad Vet Puzzle: (from The Internet, 1998)
A mad veterinarian has created three animal transmogrifying
machines.
Place a cat in the input bin of the first machine, press the
button, and whirr... bing! Open the output bins to find two
dogs and five mice.
The second machine can convert a dog into three cats and
three mice.
The third machine can convert a mouse into a cat and a dog.
Each machine can also operate in reverse (e.g. if you’ve got two
dogs and five mice, you can convert them into a cat).
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something not-as-familiar: Mad Vet Puzzles
You have one cat.
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Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something not-as-familiar: Mad Vet Puzzles
You have one cat.
1
Can you convert it into seven mice?
2
Can you convert it into a pack of dogs, with no mice or cats
left over?
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something not-as-familiar: Mad Vet Puzzles
A site for more info about Mad Vet Puzzles (The ’Mad Bob’ site):
Google:
Mad Bob’s Mad Vet Puzzles
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something not-as-familiar: Mad Vet Puzzles
A site for more info about Mad Vet Puzzles (The ’Mad Bob’ site):
Google:
Mad Bob’s Mad Vet Puzzles
A New York Times Puzzle Blog:
Google:
Numberplay: The Mad Veterinarian
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Something not-as-familiar: Mad Vet Puzzles
A site for more info about Mad Vet Puzzles (The ’Mad Bob’ site):
Google:
Mad Bob’s Mad Vet Puzzles
A New York Times Puzzle Blog:
Google:
Numberplay: The Mad Veterinarian
Mad Vet Puzzles are NOT as well-studied as the Fibonacci Puzzle.
But (surprisingly?) there is a nice connection between them !
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet scenarios
A Mad Vet scenario is a situation such as the one Mad Bob
constructed.
We assume:
1. Each species is paired up with a machine;
2. Each machine can also operate in reverse; and
3. Each machine is “one to some”
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Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
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Mat Vet Scenario #1
Scenario #1. Suppose a Mad Veterinarian has three machines
with the following properties.
Machine 1 turns one Ant into one Beaver;
Machine 2 turns one Beaver into one Ant, one Beaver and one
Cougar;
Machine 3 turns one Cougar into one Ant and one Beaver.
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mat Vet Scenario #1
Scenario #1. Suppose a Mad Veterinarian has three machines
with the following properties.
Machine 1 turns one Ant into one Beaver;
Machine 2 turns one Beaver into one Ant, one Beaver and one
Cougar;
Machine 3 turns one Cougar into one Ant and one Beaver.
So, for example,
A
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mat Vet Scenario #1
Scenario #1. Suppose a Mad Veterinarian has three machines
with the following properties.
Machine 1 turns one Ant into one Beaver;
Machine 2 turns one Beaver into one Ant, one Beaver and one
Cougar;
Machine 3 turns one Cougar into one Ant and one Beaver.
So, for example,
A⇒B
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mat Vet Scenario #1
Scenario #1. Suppose a Mad Veterinarian has three machines
with the following properties.
Machine 1 turns one Ant into one Beaver;
Machine 2 turns one Beaver into one Ant, one Beaver and one
Cougar;
Machine 3 turns one Cougar into one Ant and one Beaver.
So, for example,
A ⇒ B ⇒ A, B, C
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mat Vet Scenario #1
Scenario #1. Suppose a Mad Veterinarian has three machines
with the following properties.
Machine 1 turns one Ant into one Beaver;
Machine 2 turns one Beaver into one Ant, one Beaver and one
Cougar;
Machine 3 turns one Cougar into one Ant and one Beaver.
So, for example,
A ⇒ B ⇒ A, B, C ⇒ 2A, 2B
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mat Vet Scenario #1
Scenario #1. Suppose a Mad Veterinarian has three machines
with the following properties.
Machine 1 turns one Ant into one Beaver;
Machine 2 turns one Beaver into one Ant, one Beaver and one
Cougar;
Machine 3 turns one Cougar into one Ant and one Beaver.
So, for example,
A ⇒ B ⇒ A, B, C ⇒ 2A, 2B ⇒ 3A, B
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mat Vet Scenario #1
Scenario #1. Suppose a Mad Veterinarian has three machines
with the following properties.
Machine 1 turns one Ant into one Beaver;
Machine 2 turns one Beaver into one Ant, one Beaver and one
Cougar;
Machine 3 turns one Cougar into one Ant and one Beaver.
So, for example,
A ⇒ B ⇒ A, B, C ⇒ 2A, 2B ⇒ 3A, B ⇒ 4A
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
From Mad Vet Scenarios to graphs
Given any Mad Vet scenario, its corresponding Mad Vet graph is:
a drawing (“directed graph”),
consisting of points and arrows (“vertices” and “edges”),
which gives the info about what’s going on with the machines.
(We only draw the “forward” direction of the machines.)
Example. Mad Vet scenario #1 has the following Mad Vet graph.
A
>• j
•C o
Recall:
Machine 1: A → B,
B
8 •Q
Machine 2: B → A, B, C,
Machine 3: C → A,B
UCCS
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Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
... and vice versa: from graphs to Mad Vet Scenarios
Example: Consider this directed graph:
•D u e
•v q
% w
1•
This graph would describe a Mad Vet Scenario with three species:
Urchins, Vermin, Warthogs
Machine 1: Urchin → Vermin, Warthog
Machine 2: Vermin → Urchin, Warthog
Machine 3: Warthog → Urchin, Vermin
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet equivalence
Some notation: Let’s say there are n different species. Choose
some “order” to list them.
E.g., in Scenario #1, first list Ants, then Beavers, then Cougars.
Then any collection of animals corresponds to some n-vector, with
entries taken from the set {0, 1, 2, . . .}.
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet equivalence
Some notation: Let’s say there are n different species. Choose
some “order” to list them.
E.g., in Scenario #1, first list Ants, then Beavers, then Cougars.
Then any collection of animals corresponds to some n-vector, with
entries taken from the set {0, 1, 2, . . .}.
For instance, in Scenario #1 a collection of two Beavers and five
Cougars would correspond to (0, 2, 5).
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet equivalence
Some notation: Let’s say there are n different species. Choose
some “order” to list them.
E.g., in Scenario #1, first list Ants, then Beavers, then Cougars.
Then any collection of animals corresponds to some n-vector, with
entries taken from the set {0, 1, 2, . . .}.
For instance, in Scenario #1 a collection of two Beavers and five
Cougars would correspond to (0, 2, 5).
We agree that an “empty” collection of animals is not of interest
here. In other words, the vector (0, 0, . . . , 0) is not allowed.
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Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
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Mad Vet equivalence
There is a naturally arising relation ∼ on these vectors:
Given a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ) , we write
a∼b
if there is a sequence of Mad Vet machine moves that will change
the collection of animals associated with vector a into the
collection of animals associated with vector b.
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Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet equivalence
There is a naturally arising relation ∼ on these vectors:
Given a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ) , we write
a∼b
if there is a sequence of Mad Vet machine moves that will change
the collection of animals associated with vector a into the
collection of animals associated with vector b.
(Aside: Using the three properties of a Mad Vet scenario, it is
straightforward to show that ∼ is an equivalence relation.)
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Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
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Mad Vet equivalence
Example. Suppose that our Mad Vet of Scenario #1 starts with
one Ant; in other words, with (1, 0, 0).
(Recall:
Machine 1: A → B
Machine 2: B → A, B, C
Machine 3: C → A,B)
Then, rewriting in this new notation what we’ve already seen,
(1, 0, 0) ∼ (0, 1, 0) ∼ (1, 1, 1) ∼ (2, 2, 0) ∼ (3, 1, 0) ∼ (4, 0, 0).
As a result, (1, 0, 0) ∼ (4, 0, 0).
And (4, 0, 0) ∼ (1, 0, 0) too ...
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Introduction and brief history
Mad Vet groups
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Mad Vet equivalence
General math idea: There are many situations where different
mathematical symbols stand for the same quantity:
Fractions:
3
6
means the same as 12 , ...
Clock arithmetic: 3 (mod12) means the same as 15 (mod12), ...
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet equivalence
General math idea: There are many situations where different
mathematical symbols stand for the same quantity:
Fractions:
3
6
means the same as 12 , ...
Clock arithmetic: 3 (mod12) means the same as 15 (mod12), ...
In the context of Mad Vet Puzzles, we agree that different vectors
stand for the SAME collection (of animals) if we can get from one
of the vectors to the other by some sequence of machines.
E.g., we agree that (1, 0, 0) stands for the same collection as
(4, 0, 0) in Scenario #1.
( and as (0, 1, 0), and as (1, 1, 1), and as (2, 2, 0), and as (3, 1, 0), . . . )
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Gene Abrams, UCCS
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Mad Vet groups
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Mad Vet equivalence
Some notation: We put a vector in brackets to denote the set of
all the vectors which are the SAME as the given one in the Mad
Vet’s office.
So for Scenario #1 we could write
[(1, 0, 0)] = [(0, 1, 0)] = [(1, 1, 1)] = [(2, 2, 0)] = [(3, 1, 0)] = [(4, 0, 0)] = · · · .
Also, we could write, for example
[(2, 0, 0)] = [(1, 1, 0)] = [(2, 1, 1)] = · · · .
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Mad Vet groups
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Mad Vet equivalence
(Recall:
Machine 1: A → B
Machine 2: B → A, B, C
Machine 3: C → A,B)
Claim. In Scenario #1, there are exactly three different “bracket
vectors” of animals:
{ [(1, 0, 0)], [(2, 0, 0)], [(3, 0, 0)] }.
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Mad Vet groups
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Mad Vet equivalence
(Recall:
Machine 1: A → B
Machine 2: B → A, B, C
Machine 3: C → A,B)
Claim. In Scenario #1, there are exactly three different “bracket
vectors” of animals:
{ [(1, 0, 0)], [(2, 0, 0)], [(3, 0, 0)] }.
Reason. It’s not hard to see that any [(a, b, c)] is equivalent to
one of [(1, 0, 0)], [(2, 0, 0)], or [(3, 0, 0)].
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Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet equivalence
(Recall:
Machine 1: A → B
Machine 2: B → A, B, C
Machine 3: C → A,B)
Claim. In Scenario #1, there are exactly three different “bracket
vectors” of animals:
{ [(1, 0, 0)], [(2, 0, 0)], [(3, 0, 0)] }.
Reason. It’s not hard to see that any [(a, b, c)] is equivalent to
one of [(1, 0, 0)], [(2, 0, 0)], or [(3, 0, 0)].
Showing that these three brackets are different takes some
(straightforward) work; let’s not do that today. (This question is
similar to the Mad Bob question!)
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Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
1 Introduction and brief history
2 Mad Vet groups
3 Here’s where Fibonacci comes in ...
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Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Semigroups, monoids, and groups
Reminder / review of notation.
1
semigroup:
associative operation.
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Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Semigroups, monoids, and groups
Reminder / review of notation.
1
semigroup: associative operation.
e.g., N = {1, 2, 3, ...} under addition.
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Semigroups, monoids, and groups
Reminder / review of notation.
1
semigroup: associative operation.
e.g., N = {1, 2, 3, ...} under addition.
2
monoid:
semigroup, with an identity element.
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Semigroups, monoids, and groups
Reminder / review of notation.
1
semigroup: associative operation.
e.g., N = {1, 2, 3, ...} under addition.
2
monoid: semigroup, with an identity element.
e.g., Z+ = {0, 1, 2, 3, ...} under addition.
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Semigroups, monoids, and groups
Reminder / review of notation.
1
semigroup: associative operation.
e.g., N = {1, 2, 3, ...} under addition.
2
monoid: semigroup, with an identity element.
e.g., Z+ = {0, 1, 2, 3, ...} under addition.
3
group:
monoid, for which each element has an inverse.
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Semigroups, monoids, and groups
Reminder / review of notation.
1
semigroup: associative operation.
e.g., N = {1, 2, 3, ...} under addition.
2
monoid: semigroup, with an identity element.
e.g., Z+ = {0, 1, 2, 3, ...} under addition.
3
group: monoid, for which each element has an inverse.
e.g., Z = {−3, −2, −1, 0, 1, 2, 3, ...} under addition.
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Semigroups, monoids, and groups
Reminder / review of notation.
1
semigroup: associative operation.
e.g., N = {1, 2, 3, ...} under addition.
2
monoid: semigroup, with an identity element.
e.g., Z+ = {0, 1, 2, 3, ...} under addition.
3
group: monoid, for which each element has an inverse.
e.g., Z = {−3, −2, −1, 0, 1, 2, 3, ...} under addition.
e.g., for any positive integer n, the “clock arithmetic group”
Zn = {1, 2, . . . , n}, under addition mod n.
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Semigroups, monoids, and groups
Reminder / review of notation.
1
semigroup: associative operation.
e.g., N = {1, 2, 3, ...} under addition.
2
monoid: semigroup, with an identity element.
e.g., Z+ = {0, 1, 2, 3, ...} under addition.
3
group: monoid, for which each element has an inverse.
e.g., Z = {−3, −2, −1, 0, 1, 2, 3, ...} under addition.
e.g., for any positive integer n, the “clock arithmetic group”
Zn = {1, 2, . . . , n}, under addition mod n.
e.g., for any positive integers m, n, the “direct product” (i.e.,
ordered pairs) Zm × Zn .
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Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet semigroups
Start with a Mad Vet scenario. Define an addition process on
bracket vectors:
[x] + [y ] = [x + y ].
Interpret as “unions” of collections of animals.
Example.
(Scenario #1:
M 1: A → B
M 2: B → A, B, C
M 3: C → A,B)
The bracket vectors are {[(1, 0, 0)], [(2, 0, 0)], [(3, 0, 0)]}.
We get, for instance,
[(1, 0, 0)] + [(1, 0, 0)] = [(1 + 1, 0, 0)] = [(2, 0, 0)],
as we’d expect.
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet semigroups
Start with a Mad Vet scenario. Define an addition process on
bracket vectors:
[x] + [y ] = [x + y ].
Interpret as “unions” of collections of animals.
Example.
(Scenario #1:
M 1: A → B
M 2: B → A, B, C
M 3: C → A,B)
The bracket vectors are {[(1, 0, 0)], [(2, 0, 0)], [(3, 0, 0)]}.
We get, for instance,
[(1, 0, 0)] + [(1, 0, 0)] = [(1 + 1, 0, 0)] = [(2, 0, 0)],
as we’d expect. But also
[(1, 0, 0)] + [(3, 0, 0)] = [(4, 0, 0)]
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet semigroups
Start with a Mad Vet scenario. Define an addition process on
bracket vectors:
[x] + [y ] = [x + y ].
Interpret as “unions” of collections of animals.
Example.
(Scenario #1:
M 1: A → B
M 2: B → A, B, C
M 3: C → A,B)
The bracket vectors are {[(1, 0, 0)], [(2, 0, 0)], [(3, 0, 0)]}.
We get, for instance,
[(1, 0, 0)] + [(1, 0, 0)] = [(1 + 1, 0, 0)] = [(2, 0, 0)],
as we’d expect. But also
[(1, 0, 0)] + [(3, 0, 0)] = [(4, 0, 0)] = [(1, 0, 0)].
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet semigroups
Start with a Mad Vet scenario. Define an addition process on
bracket vectors:
[x] + [y ] = [x + y ].
Interpret as “unions” of collections of animals.
Example.
(Scenario #1:
M 1: A → B
M 2: B → A, B, C
M 3: C → A,B)
The bracket vectors are {[(1, 0, 0)], [(2, 0, 0)], [(3, 0, 0)]}.
We get, for instance,
[(1, 0, 0)] + [(1, 0, 0)] = [(1 + 1, 0, 0)] = [(2, 0, 0)],
as we’d expect. But also
[(1, 0, 0)] + [(3, 0, 0)] = [(4, 0, 0)] = [(1, 0, 0)].
So [(3, 0, 0)] behaves like an identity element w/resp to [(1, 0, 0)]
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Mad Vet groups
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Mad Vet semigroups
Similarly
[(2, 0, 0)]+[(3, 0, 0)] = [(2, 0, 0)], and [(3, 0, 0)]+[(3, 0, 0)] = [(3, 0, 0)].
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Mad Vet groups
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Mad Vet semigroups
Similarly
[(2, 0, 0)]+[(3, 0, 0)] = [(2, 0, 0)], and [(3, 0, 0)]+[(3, 0, 0)] = [(3, 0, 0)].
So for this Mad Vet scenario the Mad Vet semigroup is a monoid,
with identity [(3, 0, 0)].
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Mad Vet groups
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Mad Vet semigroups
Actually, since [(1, 0, 0)] + [(2, 0, 0)] = [(3, 0, 0)], each of the three
elements has an inverse.
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Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet semigroups
Actually, since [(1, 0, 0)] + [(2, 0, 0)] = [(3, 0, 0)], each of the three
elements has an inverse.
So the set of three bracket vectors for this Mad Vet Scenario forms
a group,
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Mad Vet groups
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Mad Vet semigroups
Actually, since [(1, 0, 0)] + [(2, 0, 0)] = [(3, 0, 0)], each of the three
elements has an inverse.
So the set of three bracket vectors for this Mad Vet Scenario forms
a group, necessarily Z3 .
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Mad Vet groups
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Mad Vet semigroups
Actually, since [(1, 0, 0)] + [(2, 0, 0)] = [(3, 0, 0)], each of the three
elements has an inverse.
So the set of three bracket vectors for this Mad Vet Scenario forms
a group, necessarily Z3 .
Notation remark:
Sometimes we write [A] for [(1, 0, 0)], [B] for [(0, 1, 0)], etc ...
So, e.g., the three bracket vectors in Scenario #1 consist of the set
{[A], 2[A], 3[A]}.
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Mad Vet groups
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Another Mad Vet Scenario (Scenario #2)
Here’s a Mad Vet with a different set of machines:
?A k
Co
So:
Machine 1: A → B,C
9B
Machine 2: B → A, C
Machine 3: C → A,B)
What are the Mad Vet bracket vectors here?
{ [(1, 0, 0)], [(0, 1, 0)], [(0, 0, 1)], [(1, 1, 1)] }
Or, in the other notation, { [A], [B], [C ], [A] + [B] + [C ] }.
Turns out: these also form a group, Z2 × Z2 .
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Mad Vet groups
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Mad Vet semigroups
There are Mad Vet Scenarios where the bracket vectors for that
scenario do NOT form a group. For instance, the Mad Vet
Scenario for this graph.
?A
Co
B
Here the bracket vectors behave like the set N = {1, 2, 3, ...}.
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Mad Vet groups
A Big Question:
Given a Mad Vet scenario,
when does the set of bracket vectors form a group?
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Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet groups
A Big Question:
Given a Mad Vet scenario,
when does the set of bracket vectors form a group?
A Big Answer:
Look at the Mad Vet graph, call it Γ.
- If Γ contains at least one cycle, and
- if you can walk from every vertex in Γ to every cycle in Γ by a
sequence of edges, and
- if Γ isn’t just a ’basic cycle’,
then the bracket vectors form a group.
And vice versa.
“Mad Vet group” of Γ
M.V.G.(Γ).
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Mad Vet groups
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Mad Vet groups
Recall the Mad Vet graphs of Scenarios #1 and #2
?A k
Γ1 =
Co
?A k
Γ2 =
9 BQ
Co
9B
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Mad Vet groups
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Mad Vet groups
Another Big Question:
When the graph has the right properties so that the bracket
vectors form a group, what group is it ????
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Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet groups
Another Big Question:
When the graph has the right properties so that the bracket
vectors form a group, what group is it ????
Another Big Answer:
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Mad Vet groups
Here’s where Fibonacci comes in ...
Mad Vet groups
Another Big Question:
When the graph has the right properties so that the bracket
vectors form a group, what group is it ????
Another Big Answer:
For today, suffice it to say that if you are given some specific graph
Γ, then it is “easy” to write code (e.g., in Mathematica) which will
tell you M.V.G.(Γ). (Matrix computations.)
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Mad Vet groups
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Mad Vet groups
For more about Mad Vet groups, see
G. Abrams and J. Sklar,
“The graph menagerie:
Abstract algebra and the Mad Veterinarian”,
Mathematics Magazine 83(3), 2010, 168-179.
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Mad Vet groups
Here’s where Fibonacci comes in ...
1 Introduction and brief history
2 Mad Vet groups
3 Here’s where Fibonacci comes in ...
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Mad Vet groups
Here’s where Fibonacci comes in ...
The “basic” cyclic graphs Cn
With the previous stuff as context, here’s a game we can play.
Take a collection of “similar” graphs for which, for each of the
graphs, the corresponding Mad Vet bracket vectors form a group.
Here’s one way we might build these. For each n ≥ 1, let Cn be
the “cycle” graph having n vertices {v1 , v2 , . . . , vn }, and n edges,
like this:
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Mad Vet groups
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The “basic” cyclic graphs Cn
C1 =
•E v1
C2 =
•I v1
•v2
•= v1
C3 =
•v3 o
•= v1
C4 =
!
•v2
•v4
!
•v2
a
•v3
}
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The bracket vectors for the basic cyclic graphs Cn aren’t so nice in
this context, because they don’t form a group.
But if we modify the Cn graphs in appropriate ways, then we get
graphs whose bracket vectors do form groups.
How can we do that?
We can add an extra edge at each vertex, in a systematic way.
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Mad Vet groups
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The graphs Cn0 :
C10 =
•E v1
C20 =
•I v1
•v2
H
C30 =
•= v1
%
•v3 o
C40 =
!
•v2
y
•= v1
!
%
•v4
•v2
a
•v3Y
y
}
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Mad Vet groups
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For each n, M.V.G.(Cn0 ) contains just one element.
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Mad Vet groups
Here’s where Fibonacci comes in ...
For each n, M.V.G.(Cn0 ) contains just one element.
Not too interesting.
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Mad Vet groups
Here’s where Fibonacci comes in ...
The graphs Cn1 :
C11 =
•E v1
C21 =
= •I
v1
}
•v2
v1
3•
=
C31 =
•v3 io
v1
3•
=
C41 =
! •v2
! •v4
•v2
U a
}
•v3 s
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Mad Vet groups
Here’s where Fibonacci comes in ...
For each n, M.V.G.(Cn1 ) is
the “clock arithmetic group” {1, 2, . . . , 2n − 1},
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For each n, M.V.G.(Cn1 ) is
the “clock arithmetic group” {1, 2, . . . , 2n − 1},
i.e.,
M.V.G.(Cn1 ) ∼
= Z2n −1 .
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Mad Vet groups
Here’s where Fibonacci comes in ...
The graphs Cn−1 :
C1−1 =
•E v1
C2−1 =
•I v1 a
! •v2
C3−1 =
C4−1 =
•= v1 k
•v3 o
!
5 •v2
•= v1 k
•v4
!
•v2
a
I
+ •v3
}
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Mad Vet groups
Here’s where Fibonacci comes in ...
The Mad Vet Group of Cn−1 for various values of n.
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Mad Vet groups
Here’s where Fibonacci comes in ...
The Mad Vet Group of Cn−1 for various values of n.
n
size of M.V.G.(Cn−1 )
1
1
2
3
3
4
4
3
5
1
6
∞
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Mad Vet groups
Here’s where Fibonacci comes in ...
The Mad Vet Group of Cn−1 for various values of n.
n
size of M.V.G.(Cn−1 )
M.V.G.(Cn−1 ) ∼
=
1
1
{0}
2
3
Z3
3
4
Z2 × Z2
4
3
Z3
5
1
{0}
6
∞
Z×Z
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Mad Vet groups
Here’s where Fibonacci comes in ...
The Mad Vet Group of Cn−1 for various values of n.
n
size of M.V.G.(Cn−1 )
M.V.G.(Cn−1 ) ∼
=
n
1
1
{0}
2
3
Z3
3
4
Z2 × Z2
4
3
Z3
5
1
{0}
6
∞
Z×Z
7
8
9
10
11
12
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Mad Vet groups
Here’s where Fibonacci comes in ...
The Mad Vet Group of Cn−1 for various values of n.
n
size of M.V.G.(Cn−1 )
M.V.G.(Cn−1 ) ∼
=
n
size of M.V.G.(Cn−1 )
M.V.G.(Cn−1 ) ∼
=
1
1
{0}
2
3
Z3
3
4
Z2 × Z2
4
3
Z3
5
1
{0}
6
∞
Z×Z
7
1
{0}
8
3
Z3
9
4
Z2 × Z2
10
3
Z3
11
1
{0}
12
∞
Z×Z
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Mad Vet groups
Here’s where Fibonacci comes in ...
The Mad Vet Group of Cn−1 for various values of n.
n
size of M.V.G.(Cn−1 )
M.V.G.(Cn−1 ) ∼
=
n
size of M.V.G.(Cn−1 )
M.V.G.(Cn−1 ) ∼
=
n
1
1
{0}
2
3
Z3
3
4
Z2 × Z2
4
3
Z3
5
1
{0}
6
∞
Z×Z
7
1
{0}
8
3
Z3
9
4
Z2 × Z2
10
3
Z3
11
1
{0}
12
∞
Z×Z
13
14
15
16
17
18
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Mad Vet groups
Here’s where Fibonacci comes in ...
The Mad Vet Group of Cn−1 for various values of n.
n
size of M.V.G.(Cn−1 )
M.V.G.(Cn−1 ) ∼
=
n
size of M.V.G.(Cn−1 )
M.V.G.(Cn−1 ) ∼
=
n
size of M.V.G.(Cn−1 )
M.V.G.(Cn−1 ) ∼
=
1
1
{0}
2
3
Z3
3
4
Z2 × Z2
4
3
Z3
5
1
{0}
6
∞
Z×Z
7
1
{0}
8
3
Z3
9
4
Z2 × Z2
10
3
Z3
11
1
{0}
12
∞
Z×Z
13
1
{0}
14
3
Z3
15
4
Z2 × Z2
16
3
Z3
17
1
{0}
18
∞
Z×Z
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Mad Vet groups
Here’s where Fibonacci comes in ...
We can prove a Theorem which says:
This pattern in the Mad Vet Groups for Cn−1 goes on forever.
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Mad Vet groups
Here’s where Fibonacci comes in ...
The graphs Cn2 :
C12 =
•E v1
C22 =
•I v1
•v2
H
•= v1 k
C32 =
•v3 o
C42 =
!
5 •v2
•= Kv1
•v4 l
, •!
v2
a
}
•v3
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Here’s where Fibonacci comes in ...
Here are two more graphs in this sequence ...
C52 =
C62 =
•= F v1
•v5 h
O
v
•v4 o
!
/ •v
2
•v3
•= F v1
!
/ •v
2
•v6
O X
•v5 o
•v3
a
}
•v4
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Mad Vet groups
Here’s where Fibonacci comes in ...
Size of the Mad Vet Group of Cn2 for various values of n.
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Mad Vet groups
Here’s where Fibonacci comes in ...
Size of the Mad Vet Group of Cn2 for various values of n.
n
1
2
3
4
5
6
7
8
9
10
11
12
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Mad Vet groups
Here’s where Fibonacci comes in ...
Size of the Mad Vet Group of Cn2 for various values of n.
n
1
2
3
4
5
6
7
8
9
10
11
12
|M.V.G.(Cn2 )|
1
1
4
5
11
16
29
45
76
121
199
320
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Mad Vet groups
Here’s where Fibonacci comes in ...
Size of the Mad Vet Group of Cn2 for various values of n.
n
1
2
3
4
5
6
7
8
9
10
11
12
|M.V.G.(Cn2 )|
1
1
4
5
11
16
29
45
76
121
199
320
Here are more values ...
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Here’s where Fibonacci comes in ...
Size of the Mad Vet Group of Cn2 for various values of n.
n
1
2
3
4
5
6
7
8
9
10
11
12
|M.V.G.(Cn2 )|
1
1
4
5
11
16
29
45
76
121
199
320
Here are more values ...
n
13
14
15
16
17
18
|M.V.G.(Cn2 )|
521
841
1364
2205
3571
5776
n
19
20
21
22
23
24
|M.V.G.(Cn2 )|
9349
15125
24476
39601
64079
103680
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Here’s where Fibonacci comes in ...
Let’s do some sample computations in, say, M.V.G.(C62 ).
[v1 ] = [v2 ] + [v3 ]
= ([v3 ] + [v4 ]) + [v3 ]
= 2[v3 ] + [v4 ]
= 2([v4 ] + [v5 ]) + [v4 ] = 3[v4 ] + 2[v5 ]
= 3([v5 ] + [v6 ]) + 2[v5 ] = 5[v5 ] + 3[v6 ]
= 5([v6 ] + [v1 ]) + 3[v6 ] = 8[v6 ] + 5[v1 ]
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So, in M.V.G.(C62 ), we get
[v1 ] = 8[v6 ] + 5[v1 ].
This gives
8[v6 ] = −4[v1 ].
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So, in M.V.G.(C62 ), we get
[v1 ] = 8[v6 ] + 5[v1 ].
This gives
8[v6 ] = −4[v1 ].
So, here, we have
F (6)[v6 ] = −(F (5) − 1)[v1 ].
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A general conclusion, and Connection #1:
Repeating ... in M.V.G.(C62 ), F (6)[v6 ] = −(F (5) − 1)[v1 ].
More generally:
Let n be any positive integer. Then, in M.V.G.(Cn2 ),
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A general conclusion, and Connection #1:
Repeating ... in M.V.G.(C62 ), F (6)[v6 ] = −(F (5) − 1)[v1 ].
More generally:
Let n be any positive integer. Then, in M.V.G.(Cn2 ),
F (n)[vn ] =
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A general conclusion, and Connection #1:
Repeating ... in M.V.G.(C62 ), F (6)[v6 ] = −(F (5) − 1)[v1 ].
More generally:
Let n be any positive integer. Then, in M.V.G.(Cn2 ),
F (n)[vn ] = − (F (n−1) − 1)[v1 ].
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Mad Vet groups
Here’s where Fibonacci comes in ...
A general conclusion, and Connection #1:
Repeating ... in M.V.G.(C62 ), F (6)[v6 ] = −(F (5) − 1)[v1 ].
More generally:
Let n be any positive integer. Then, in M.V.G.(Cn2 ),
F (n)[vn ] = − (F (n−1) − 1)[v1 ].
So Fibonacci’s rabbits and the Mad Veterinarian
are connected !!
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Connection #2:
Notation: Denote the size of M.V.G(Cn2 ) by H2 (n).
Recall the sizes of the Mad Vet groups of the Cn2 graphs (1 ≤ n ≤ 12):
n
1
2
3
4
5
6
7
8
9
10
11
12
H2 (n)
1
1
4
5
11
16
29
45
76
121
199
320
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Mad Vet groups
Here’s where Fibonacci comes in ...
Connection #2:
Notation: Denote the size of M.V.G(Cn2 ) by H2 (n).
Recall the sizes of the Mad Vet groups of the Cn2 graphs (1 ≤ n ≤ 12):
n
1
2
3
4
5
6
7
8
9
10
11
12
H2 (n)
1
1
4
5
11
16
29
45
76
121
199
320
Proposition: For all n ≥ 3,
(
H2 (n − 1) + H2 (n − 2)
H2 (n) =
H2 (n − 1) + H2 (n − 2) + 2
if n is even
if n is odd
So the H2 sequence is “Fibonacci-ish”.
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Digression
Digression: The Online Encyclopedia of Integer Sequences
google:
OEIS
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Digression
Digression: The Online Encyclopedia of Integer Sequences
google:
OEIS
Digression: C.B. Haselgrove
google:
Haselgrove mathematics
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Digression
Digression: The Online Encyclopedia of Integer Sequences
google:
OEIS
Digression: C.B. Haselgrove
google:
Haselgrove mathematics
A note on Fermat’s Last Theorem and the Mersenne Numbers,
in: Eureka: the Archimedeans’ Journal, vol. 11, 1949, pp 19-22.
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Connection #3
Now let’s look at the Fibonacci sequence and the H2 sequence
side-by-side:
n
1
2
3
4
5
6
7
8
9
10
11
12
F (n)
H2 (n)
1
1
1
1
2
4
3
5
5
11
8
16
13
29
21
45
34
76
55
121
89
199
144
320
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Connection #3
Now let’s look at the Fibonacci sequence and the H2 sequence
side-by-side:
n
1
2
3
4
5
6
7
8
9
10
11
12
F (n)
H2 (n)
1
1
1
1
2
4
3
5
5
11
8
16
13
29
21
45
34
76
55
121
89
199
144
320
Proposition: For all n ≥ 2,
(
F (n − 1) + F (n + 1) − 2 if n is even
H2 (n) =
F (n − 1) + F (n + 1)
if n is odd
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Gene Abrams, UCCS
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Connection #4
Along the way, the following numbers turn out to be of great
interest. For each n ≥ 2, define
d(n) = g.c.d.(F (n), F (n−1) − 1)
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Connection #4
Along the way, the following numbers turn out to be of great
interest. For each n ≥ 2, define
d(n) = g.c.d.(F (n), F (n−1) − 1)
What are these numbers?
(reminder: g.c.d.(0, m) = m for any positive integer m)
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Connection #4
Along the way, the following numbers turn out to be of great
interest. For each n ≥ 2, define
d(n) = g.c.d.(F (n), F (n−1) − 1)
What are these numbers?
(reminder: g.c.d.(0, m) = m for any positive integer m)
n
1
2
3
4
5
6
7
8
9
10
11
12
F (n)
d(n)
1
1
1
1
2
2
3
1
5
1
8
4
13
1
21
3
34
2
55
11
89
1
144
8
···
···
···
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Introduction and brief history
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Connection #4
The d(n) sequence had arisen in another context;
formula was given for it ... (see O.E.I.S.)
and no explicit
n
1
2
3
4
5
6
7
8
9
10
11
12
F (n)
d(n)
1
1
1
1
2
2
3
1
5
1
8
4
13
1
21
3
34
2
55
11
89
1
144
8
···
···
···
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Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Connection #4
The d(n) sequence had arisen in another context;
formula was given for it ... (see O.E.I.S.)
and no explicit
n
1
2
3
4
5
6
7
8
9
10
11
12
F (n)
d(n)
1
1
1
1
2
2
3
1
5
1
8
4
13
1
21
3
34
2
55
11
89
1
144
8
Proposition: For any positive integer m,
(
1 if 2m + 1 ≡ 1
d(2m + 1) =
2 if 2m + 1 ≡ 3
(
F (m) + F (m + 2)
d(2m + 2) =
F (m + 1)
···
···
···
or 5 mod 6
mod 6
if m is even
if m is odd
So we have an explicit formula for d(n) for all integers n ≥ 1.
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Connection #5
Lemma: (d(n))2 divides H2 (n) for all n.
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Connection #5
Lemma: (d(n))2 divides H2 (n) for all n.
Theorem: For any integer n,
M.V.G.(Cn2 ) ∼
= Zd(n) × Z H2 (n) .
d(n)
In particular, M.V.G.(Cn2 ) is cyclic precisely when d(n) = 1,
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Connection #5
Lemma: (d(n))2 divides H2 (n) for all n.
Theorem: For any integer n,
M.V.G.(Cn2 ) ∼
= Zd(n) × Z H2 (n) .
d(n)
In particular, M.V.G.(Cn2 ) is cyclic precisely when d(n) = 1,
so precisely when d = 2, or d = 4, or d ≡ 1 or 5 mod 6.
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Gene Abrams, UCCS
Introduction and brief history
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Connection #6
So ...
Who Cares?
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Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
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Here’s where Fibonacci comes in ...
Connection #6
So ...
Who Cares?
For any directed graph E and field K , we can form the “Leavitt
path algebra of E with coefficients in K .” LK (E )
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Gene Abrams, UCCS
Introduction and brief history
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Here’s where Fibonacci comes in ...
Connection #6
So ...
Who Cares?
For any directed graph E and field K , we can form the “Leavitt
path algebra of E with coefficients in K .” LK (E )
Connection:
K0 (LK (E )) ∼
= M.V.G.(E ).
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Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Connection #6
So ...
Who Cares?
For any directed graph E and field K , we can form the “Leavitt
path algebra of E with coefficients in K .” LK (E )
Connection:
K0 (LK (E )) ∼
= M.V.G.(E ).
And we can use the description of M.V.G.(E ) (plus some other
stuff) to get information about the structure of LK (E ).
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Connection #6
So ...
Who Cares?
For any directed graph E and field K , we can form the “Leavitt
path algebra of E with coefficients in K .” LK (E )
Connection:
K0 (LK (E )) ∼
= M.V.G.(E ).
And we can use the description of M.V.G.(E ) (plus some other
stuff) to get information about the structure of LK (E ).
In particular, knowing the structure of M.V.G.(Cn2 ) gives really
nice information about LK (Cn2 ).
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Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
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Here’s where Fibonacci comes in ...
What’s next?
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
What’s next?
Can we describe M.V.G.(Cn3 ) ??
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
What’s next?
Can we describe M.V.G.(Cn3 ) ??
Can we describe M.V.G.(Cn−2 ) ??
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Questions?
Questions?
UCCS
Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS
Introduction and brief history
Mad Vet groups
Here’s where Fibonacci comes in ...
Questions?
Questions?
Thank you.
Thanks to Simons Foundation.
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Fibonacci’s rabbits visit the Mad Veterinarian
Gene Abrams, UCCS